Two conjugate transpose datuming algorithms

We can epitomize the datuming algorithm in a depth varying velocity by writing the successive operators in matrix form. For simplicity I will take only a datum with three depth levels similar to the one in Figure . The algorithm can be generalized for any number of depth levels and any datum geometry. For each value of $\omega$ we have  

$$\displaystyle{ \left[ \begin{array} {ccc} A&B&C \end{arra... ... \left[ \begin{array} {c} P(x,z=0,{\omega})\end{array}\right] }$$ (6)

where $P(x,z=0,\omega)$ represents the zero-offset data after Fourier transformation along the time axis. The matrix

$$\left[ \begin{array} {c} FT \\ \end{array}\right]$$

performs the Fourier Transform matrix of the horizontal space variable. Therefore the matrix

$$\left[ \begin{array} {ccc} FT^* & & \\ &FT^* & \\ & &FT^* \\ \end{array}\right]$$

is composed of inverse Fourier Transform block matrices, each being the conjugate transpose to the direct transformation. The matrix

$$\left[ \begin{array} {c} W_1\\ W_2 W_1\\ W_3 W_2 W_1\\ \end{array}\right]$$

contains three blocks corresponding to the product of diagonal matrices of the form  

$$W_i= \left[ \begin{array} {cccccc} e^{ik_{zi1} \Delta z} &... ...& & \\ & & & & &e^{ik_{zi6} \Delta z} \\ \end{array}\right] .$$ (7)

On each diagonal are the phase-shifting exponentials necessary to upward extrapolate the wavefield to a depth level. The value of (k_z)_ij is given by the dispersion relation:

$${(k_z)}_{ij}={\sqrt{{{\omega^2} \over v(z_i)^2}-k^2_{xj}}}.$$

 

Datumatrix Figure 2 Datuming for only three depth levels. Data extrapolated to the first depth level is extracted by matrix A, to the second level by matrix B and to the third by matrix C.

The matrices A,B,C are matrices corresponding to the shape of the datum. For the case shown in Figure the matrices A,B,C are

$$A= \left[ \begin{array} {cccccc} 1& & & & & \\ &1& & & & \\... ... & & &.& & \\ & & & &1& \\ & & & & &1 \\ \end{array}. \right]$$

The conjugate transpose algorithm can be obtained by reversing the order of matrix multiplication in equation (6) and transposing-conjugating each matrix. The matrices A,B,C are diagonal matrices and therefore equal to the transpose. For each value of the frequency $\omega$ the conjugate datuming is  

$$\displaystyle{ \left[ \begin{array} {c} FT^* \\ \end{arra... ...ft[ \begin{array} {c} P(x,z_{dat},{\omega})\end{array}\right] }$$ (8)

where $P(x,z_{dat},\omega)$ represents the wavefield recorded on the topographic datum. The matrices W^*_i are conjugate transposed of the diagonal matrices W_i which are explicitly represented in equation (7).